what degree is this polynomial function? a second b third c fourth d fifth

Explanation:

Step1: Analyze end - behavior and turning points

To determine the degree of a polynomial from its graph, we can use two main features: end - behavior and the number of turning points. The degree of a polynomial (n) and the number of turning points (T) are related by the inequality (T\leq n - 1). Also, the end - behavior of a polynomial is determined by the leading term (a_nx^n). If (n) is odd, the ends of the graph go in opposite directions (one up and one down), and if (n) is even, the ends go in the same direction.

Looking at the graph:

• The left - hand end of the graph (as (x\rightarrow-\infty)) is going down (towards (-\infty)) and the right - hand end (as (x\rightarrow\infty)) is going up (towards (\infty)). This indicates that the degree of the polynomial is odd (since for odd (n), (\lim_{x\rightarrow-\infty}a_nx^n=-\infty) and (\lim_{x\rightarrow\infty}a_nx^n=\infty) when (a_n>0), or vice - versa for (a_n < 0)). So we can eliminate the even - degree options (second and fourth degree, options A and C).
• Now, let's consider the number of turning points. A turning point is a point where the graph changes from increasing to decreasing or vice - versa. From the graph, we can see that there are 2 turning points. Using the formula (T\leq n - 1), if (T = 2), then (n-1\geq2), so (n\geq3).

Step2: Distinguish between third and fifth degree

A third - degree polynomial ((n = 3)) has a maximum of (n - 1=2) turning points. A fifth - degree polynomial ((n = 5)) has a maximum of (n - 1 = 4) turning points. Since our graph has 2 turning points, and we already know the degree is odd, a third - degree polynomial is consistent with having 2 turning points. A fifth - degree polynomial would be expected to have more turning points (up to 4) if it is a general fifth - degree polynomial. Also, the end - behavior of a fifth - degree polynomial is also odd - degree (opposite ends), but the number of turning points here is too few for a fifth - degree polynomial (since a fifth - degree polynomial can have up to 4 turning points, but our graph has only 2). So the polynomial is of third degree.

Answer:

B. third